Here is the full list of all 31 Sid Meier's Starships achievements.
Win the game as Harmony
Win the game as Supremacy
Win a mission by bringing your flagship to a warp portal
Win a Science Victory
Win a Wonder Victory
Research 5 techs to level 4
Control 5 Galactic Wonders
Upgrade the stealth module to level 6
Raise a planetary improvement to level 5
Win the game as Purity
Win a Population Victory
Take over an enemy's home planet
Increase the population on a planet to 10
Have 4 or more fighters on the map at once during a battle
Research a tech to level 4
Destroy a ship with a torpedo
Upgrade the lasers on your ship
Create a warp portal between two planets
Score a critical hit on an enemy ship
Bring a planet fully into your empire
Use a jump gate during battle
Repair a critical hit on a ship in your fleet
Win a mission by bringing the colony ship to a warp portal
Have a ship in your fleet move 7 or more hexs in one turn during a battle
Upgrade the population on an earth-like planet
Travel to another planet
Defend your home planet
Give away a tech through negotiating with an enemy
Defeat the Pirate planet
Start a game using a save game from Beyond Earth
Win a Domination Victory